I am wondering what portions of the New Testament are purported to have originally been written in Aramaic, and by whom. I am aware that Seventh Day Adventist and Nestorian scholars often assert Aramaic primacy, but they generally assert that the entire New Testament was originally written in Aramaic. I am interested in scholarly (non-doctrinal, as far as possible) literature that would support Aramaic primacy in specific portions of the New Testament. I am aware of the Q theory, but I am looking for those who claim that entire books or even the entire New Testament was originally written in Aramaic.

I'm not aware of any significant portion of the New Testament to have been written in Aramaic, but a sizable chunk of the Book of Daniel is (for literary and stylistic purposes).
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Jamin GreyMar 6 '13 at 2:30

OTOH, There was a proto gospel, called variously as “The Gospel of the Hebrews” or “The Gospel of the Eabionites” or the “Gospel of the Nazorians” that were very close to Matthew, but lacking the first two chapters.
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Robert Rey BlackOct 17 '13 at 18:23

2 Answers
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The oldest and most studied claim of this sort is that there was an Aramaic gospel which served as a source for the synoptics. In particular, many early church fathers believed that the original version of Matthew was written in Aramaic based in part on the writings of Papias who said: "For Matthew composed the logia [sayings] in Hebrew style; but each recorded them as he was able." This claim is less popular now, because most scholars think that Mark was the first gospel written, not Matthew, nonetheless it is a venerable claim with some decent evidence behind it. (Note that this is quite different from Q, Q requires Markan priority, while Aramaic original of Matthew requires Matthean priority.)

Similarly, several ancient authors argued that the book of Hebrews was a letter written by Paul in Aramaic to Jews, and that the Greek version was a translation made by Luke (explaining why the Greek style is different and superior to Paul's other writings).

More modern scholars have identified smaller snippets which they think were translated from Aramaic sources. In particular, many individual sayings of Jesus are sometimes thought to be translated from Aramaic, though it's a bit hard to be sure on any particular saying. Another example is the creed in Romans 1:3-4.

Check out James R. Edwards' book, The Hebrew Gospel and the Development of the Synoptic Tradition. From my review in the Stone-Campbell Journal, 14:2 Fall, 2011:

This volume presents an older idea concerning the formation of the Synoptic Gospels, while connecting the data in a new way. The current volume resurrects an older idea concerning the formation of the Synoptic Gospels, which had been laid aside in favor of theories that relied either on Markan priority and “Q,” or Markan posterity. This older theory held that Matthew was the first Gospel and written originally in Hebrew and not Greek. This theory was discounted by most NT scholars in the earlier days of Gospel criticism partly due to a preference for Greek originals and partly due to the lack of an extant copy of the Hebrew Gospel. The idea of a Hebrew Gospel received much criticism by later NT scholars because the canonical Greek Matthew was shown to be a composition originally in Greek and not to be a Greek translation of an earlier Hebrew document. Combine these factors with anti-Semitism in Germany in the 1800s and 1900s, and one can appreciate why Hebrew Gospel theories did not receive much scholarly attention. Edwards reexamines the data and connects the data differently to illustrate how the Hebrew Gospel is a source for Luke and not Matthew.

This volume is laid out in eight chapters and includes an introduction, an epilogue, and three appendices. Edwards begins by citing and discussing various references to the Hebrew Gospel in the first nine centuries. Full text citations for these references are found in the first appendix (263-291). Next, he treats actual quotations from the Hebrew Gospel. The full text of these citations is found in the footnotes of the second chapter (44-96). Then he defines the Hebrew Gospel in the third chapter. In the fourth chapter, Edwards directs the reader to the Gospel of Luke, where he treats the Semitisms in that gospel. The full text of these citations is found in the second and third appendices (292-335). He decides
that Luke’s source must be a Hebrew source and not an Aramaic source in the fifth chapter. Edwards’s list of nine classifications of Hebraisms in Luke’s gospel will convince students of the Hebrew language that Luke’s source is not Aramaic (185). The last three chapters explain why this theory has been neglected by most NT scholars, discuss sources for “double tradition” material, and relate the Hebrew Gospel to the canonical Matthew.

Edwards proposes a view that gives credence to the Patristic witness, while also giving the modern critical scholars a fair treatment. This view allows the Early Church Fathers to be taken at face value stating that Matthew came first, while illustrating how Mark is the most prior Greek document, based on modern studies. Canonical Greek Matthew came later, and its date of production is not treated by the Early Church Fathers by and large. This volume illustrates the benefit of having a NT scholar who is adept in both Hebrew and Greek.

The strength of this volume lies in the collation of quotations from and about the Hebrew Gospel and relating those to the Hebraisms of Luke. This volume “bids adieu” to “Q” (240-242), but the typical scholar will be unconvinced of the need to do this. Using the diagram provided in this volume, one might conclude that the “double tradition” is equivalent to “Q” (262), while the Hebrew Gospel might also be labeled “L.” The appendices in this volume
enable the student quick access to the data. While it would be nice to have another appendix of the actual citations from the Hebrew Gospel, these are still accessible in the footnotes of the second chapter. Interestingly, this volume does not treat Shem Tov’s Matthew, which is in Hebrew, or other Hebrew versions of Matthew. This volume is likely to spark discussion or even controversy regarding the Synoptic Gospels. No student of the formation of the Gospels should be without this up-to-date treatment of the Hebrew Gospel as a source for special Luke.

James, I found a copy of your review and copied most of it here. Please review my edit since I likely copied too much. Edwards' compromise of placing the Greek translation of Matthew after Luke is ingenious. I'll have to think on it for a while.
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Jon Ericson♦Mar 5 '13 at 22:33